Signal processing

11. The impulse response functions of four linear systems S1, S2, S3, S4 are given respectively by $${h_1}\left( t \right) = 1,$$ $${h_2}\left( t \right) = u\left( t \right),$$ $${h_3}\left( t \right) = \frac{{u\left( t \right)}}{{t + 1}},$$ $${h_4}\left( t \right) = {e^{ – 3t}}u\left( t \right)$$ Where u(t) is the unit step function. Which of these systems is time invariant, causal, and stable?

S1
S2
S3
S4

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these systems is time invariant, causal, and stable?" class="read-more button" href="https://exam.pscnotes.com/mcq/the-impulse-response-functions-of-four-linear-systems-s1-s2-s3-s4-are-given-respectively-by-h_1left-t-right-1-h_2left-t-right-uleft-t-right-h_3left-t-ri/#more-58144">Detailed SolutionThe impulse response functions of four linear systems S1, S2, S3,
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S4 are given respectively by $${h_1}\left( t \right) = 1,$$ $${h_2}\left( t \right) = u\left( t \right),$$ $${h_3}\left( t \right) = \frac{{u\left( t \right)}}{{t + 1}},$$ $${h_4}\left( t \right) = {e^{ – 3t}}u\left( t \right)$$ Where u(t) is the unit step function. Which of these systems is time invariant, causal, and stable?

13. A discrete-time all-pass system has two of its poles at 0.25<0° and 2<30°. Which one of the following statements about the system is TRUE?

It has two more poles at 0.5<30° and 4<0°
It is stable only when the impulse response is two-sided
It has constant phase response over all frequencies

Detailed SolutionA discrete-time all-pass system has two of its poles at 0.25<0° and 2<30°. Which one of the following statements about the system is TRUE?

14. A stable linear time invariant (LTI) system has a transfer function $$H\left( s \right) = {1 \over {{s^2} + s – 6}}.$$ To make this system causal it needs to be cascaded with another LTI system having a transfer function H1(s). A correct choice for H1(s) among the following options is

s + 3
s - 2
s - 6
s + 1

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system having a transfer function H1(s). A correct choice for H1(s) among the following options is" class="read-more button" href="https://exam.pscnotes.com/mcq/a-stable-linear-time-invariant-lti-system-has-a-transfer-function-hleft-s-right-1-over-s2-s-6-to-make-this-system-causal-it-needs-to-be-cascaded-with-another-lti-system-hav/#more-57354">Detailed SolutionA stable linear time invariant (LTI) system has a transfer function $$H\left( s \right) = {1 \over {{s^2} + s – 6}}.$$ To make this system causal it needs to be cascaded with another LTI system having a transfer function H1(s). A correct choice for H1(s) among the following options is

15. Assuming zero initial condition, the response y(t) of the system given below to a unit step input u(t) is \[\xrightarrow{{U\left( s \right)}}\boxed{\frac{1}{s}}\xrightarrow{{Y\left( s \right)}}\]

u(t)
tu(t)
$$rac{{{t^2}}}{2}uleft( t ight)$$
e-tu(t)

Detailed SolutionAssuming

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zero initial condition, the response y(t) of the system given below to a unit step input u(t) is \[\xrightarrow{{U\left( s \right)}}\boxed{\frac{1}{s}}\xrightarrow{{Y\left( s \right)}}\]

16. The function x(t) is shown in the figure. Even and odd parts of a unit-step function u(t) are respectively,

$$rac{1}{2},rac{1}{2}xleft( t ight)$$
$$ - rac{1}{2},rac{1}{2}xleft( t ight)$$
$$rac{1}{2}, - rac{1}{2}xleft( t ight)$$
$$ - rac{1}{2}, - rac{1}{2}xleft( t ight)$$

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d="M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z"/> Subscribe on YouTube u(t) are respectively," class="read-more button" href="https://exam.pscnotes.com/mcq/the-function-xt-is-shown-in-the-figure-even-and-odd-parts-of-a-unit-step-function-ut-are-respectively/#more-57173">Detailed SolutionThe function x(t) is shown in the figure. Even and odd parts of a unit-step function u(t) are respectively,

17. Laplace transforms of the functions tu(t) and u(t)sin(t) are respectively

$${1 over {{s^2}}},{s over {{s^2} + 1}}$$
$${1 over s},{1 over {{s^2} + 1}}$$
$${1 over {{s^2}}},{1 over {{s^2} + 1}}$$
$$s,{s over {{s^2} + 1}}$$

Detailed SolutionLaplace transforms of the functions tu(t) and u(t)sin(t) are respectively

18. It is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \[x\left[ n \right] = {c_1}\exp \left( { – \frac{{j\pi n}}{2}} \right) + {c_2}\exp \left( {\frac{{j\pi n}}{2}} \right),\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \[\xrightarrow{{x\left[ n \right]}}\boxed{\begin{array}{*{20}{c}} {n = 0} \\ \downarrow \\ {h\left[ n \right] = \left\{ {1,a,b} \right\}} \end{array}}\xrightarrow{{y\left[ n \right] = 0}}\]

a = -1, b = 1
a = 0, b = 1
a = 1, b = 1
a = 0, b = -1

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three-tap causal filter which gives zero signal as an output to and input of the form \[x\left[ n \right] = {c_1}\exp \left( { – \frac{{j\pi n}}{2}} \right) + {c_2}\exp \left( {\frac{{j\pi n}}{2}} \right),\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \[\xrightarrow{{x\left[ n \right]}}\boxed{\begin{array}{*{20}{c}} {n = 0} \\ \downarrow \\ {h\left[ n \right] = \left\{ {1,a,b} \right\}} \end{array}}\xrightarrow{{y\left[ n \right] = 0}}\]" class="read-more button" href="https://exam.pscnotes.com/mcq/it-is-desired-to-find-three-tap-causal-filter-which-gives-zero-signal-as-an-output-to-and-input-of-the-form-xleft-n-right-c_1exp-left-fracjpi-n2-right-c_2exp/#more-56949">Detailed SolutionIt is desired to find three-tap causal filter which gives zero signal as an output to and input of the form \[x\left[ n \right] = {c_1}\exp \left( { – \frac{{j\pi n}}{2}} \right) + {c_2}\exp \left( {\frac{{j\pi n}}{2}} \right),\] Where c1 and c2 are arbitrary real numbers. The desired three-tap filter is given by h[0] = 1, h[1] = a, h[2] = b and h[n] = 0 for n < 0 or n > 2. What are the values of the filter taps a and b if the output is y[n] = 0 for all n, when x[n] is as given above? \[\xrightarrow{{x\left[ n \right]}}\boxed{\begin{array}{*{20}{c}} {n = 0} \\ \downarrow \\ {h\left[ n \right] = \left\{ {1,a,b} \right\}} \end{array}}\xrightarrow{{y\left[ n \right] = 0}}\]

19. Let x[n] = x[-n]. Let X(z) be the z-transform of x[n]. If 0.5 + j0.25 is a zero of X(z), which one of the following must also be a zero of X(z).

0.5 - j0.25
$${1 over {left( {0.5 + j0.25} ight)}}$$
$${1 over {left( {0.5 - j0.25} ight)}}$$
2 + j4

Detailed

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SolutionLet x[n] = x[-n]. Let X(z) be the z-transform of x[n]. If 0.5 + j0.25 is a zero of X(z), which one of the following must also be a zero of X(z).

20. The bilateral Laplace transform of a function $$f\left( t \right) = \left\{ {\matrix{ {1,} & {{\rm{if}}\,a \le t \le b} \cr 0 & {{\rm{otherwise}}} \cr } } \right.$$ is

$${{a - b} over s}$$
$${{{e^z}left( {a - b} ight)} over s}$$
$${{{e^{ - as}} - {e^{ - bs}}} over s}$$
$${{{e^{ - left( {a - b} ight)}}} over s}$$

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{1,} & {{\rm{if}}\,a \le t \le b} \cr 0 & {{\rm{otherwise}}}
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\cr } } \right.$$ is" class="read-more button" href="https://exam.pscnotes.com/mcq/the-bilateral-laplace-transform-of-a-function-fleft-t-right-left-matrix-1-rmifa-le-t-le-b-cr-0-rmotherwise-cr-right-is/#more-56755">Detailed SolutionThe bilateral Laplace transform of a function $$f\left( t \right) = \left\{ {\matrix{ {1,} & {{\rm{if}}\,a \le t \le b} \cr 0 & {{\rm{otherwise}}} \cr } } \right.$$ is
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